
where \({\mathrm {KL}}(\nu ||\mu )\)  denotes the Kullback-Leibler divergence (REF ).
Furthermore, the minimum of \(J^{\Delta ,y}\)  is explicitly given by [1]},
\(-\log \left(\int _X e^{-\Phi ^{\Delta ,y}(u)} \, d\mu (u)\right)=\inf _{\nu \in {\mathcal {P}}(X)}J^{\Delta ,y}(\nu ).\) 
